So I've been reading the chapter of Uniform Integrability on Probability Theory by Achim Klenke which has one of the following proposition.

There is a map $h \in \mathcal{L}^{1}(\mu)$ such that $h > 0$ almost everywhere.
In the proof, it constructed increasing set $A_{n}$ with limit being the whole space $\Omega$ and each element possessing finite measure, i.e. $\mu(A_{n}) < \infty$.

I am not sure whether this is only for $\mu$ being $\sigma$-finite or others, since in the following theorem he showed result similar to the Vitali convergence theorem but without the condition that restrict set measure.